For example, adding $\ce{HCl}$ to the very insoluble $\ce{CaF2}$ will help it further dissolve. How?
I believe the concept deals with equilibrium and Le Chatelier's Principle:
$$ \ce{ CaF2 <=> Ca^2+ + 2F^-} $$
Does adding $\ce{H+}$ react with the $\ce{F-}$, which shifts the equilibrium to the right-hand side and therefore further dissolve the salt?
In general, the solubility of a sparingly soluble salt having a basic anion will increase with the addition of a strong acid. Given:
$$\ce{X+A- <--> X+ + A-}$$
where $K_{sp}$ is small and A- is a weak base, the addition of a strong acid will form HA, driving the reaction to the right and increasing $K_{sp}$.
In the specific case of
$$\ce{CaF2 <--> Ca^2+ + 2F−}$$
the addition of $\ce{HCl}$ gives
$$\ce{CaF2 + 2HCl <--> Ca^2+ + 2Cl- +2HF}$$
driving the reaction to the right, thus increasing the solubility of $\ce{CaF2}$. Note that $\ce{HF}$ itself is a weak acid, having a pKa of $3.17$, and will be in equilibrium with $\ce{H+}$ and $\ce{F-}$.
